## Linear Operators: Spectral theory |

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Page 1433

is called the indicial

roots en , . . . , en , no two of which differ by an integer , then the set of solutions of

[ * ] has a basis of the form o ; ( z ) = xosq ; ( ) , where o , is analytic and non - zero

...

is called the indicial

**equation**of [ * ] at zero . If the indicial**equation**has distinctroots en , . . . , en , no two of which differ by an integer , then the set of solutions of

[ * ] has a basis of the form o ; ( z ) = xosq ; ( ) , where o , is analytic and non - zero

...

Page 1527

The same process applied to the hypergeometric

y ; 2 ) satisfies the confluent hypergeometric

. dz2 ) This

...

The same process applied to the hypergeometric

**equation**[ 1 ] shows that @ la ,y ; 2 ) satisfies the confluent hypergeometric

**equation**( 4 ) ø + ( y - 2 ) 29 - a $ = 0. dz2 ) This

**equation**has singularities at zero and infinity . The singularity at zero...

Page 1528

The first of these algebraic

of the differential

differential

irregular ...

The first of these algebraic

**equations**, which is simply the characteristic**equation**of the differential

**equation**, is quadratic ... f being a solution of the originaldifferential

**equation**Lf = 0 , we find that L ' f ' has rational coefficients , and anirregular ...

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero